Erroneous Arrhenius: Modified Arrhenius Model Best Explains the Temperature Dependence
of Ectotherm Fitness.
Author(s): Jennifer L. Knies and Joel G. Kingsolver
Reviewed work(s):
Source: The American Naturalist, Vol. 176, No. 2 (August 2010), pp. 227-233
Published by: The University of Chicago Press for The American Society of Naturalists
Stable URL: http://www.jstor.org/stable/10.1086/653662 .
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vol. 176, no. 2
the american naturalist

august 2010
Notes and Comments
Erroneous Arrhenius: Modified Arrhenius Model Best Explains
the Temperature Dependence of Ectotherm Fitness
Jennifer L. Knies1,* and Joel G. Kingsolver 2
1. Department of Ecology and Evolutionary Biology, Box G, Brown University, Providence, Rhode Island 02912; 2. Department of
Biology, University of North Carolina, Chapel Hill, North Carolina 27599
Submitted January 10, 2010; Accepted March 26, 2010; Electronically published June 8, 2010
Online enhancements: appendix figure.
abstract: The initial rise of fitness that occurs with increasing
temperature is attributed to Arrhenius kinetics, in which rates of
reaction increase exponentially with increasing temperature. Models
based on Arrhenius typically assume single rate-limiting reactions
over some physiological temperature range for which all the ratelimiting enzymes are in 100% active conformation. We test this assumption using data sets for microbes that have measurements of
fitness (intrinsic rate of population growth) at many temperatures
and over a broad temperature range and for diverse ectotherms that
have measurements at fewer temperatures. When measurements are
available at many temperatures, strictly Arrhenius kinetics are rejected over the physiological temperature range. However, over a
narrower temperature range, we cannot reject strictly Arrhenius kinetics. The temperature range also affects estimates of the temperature dependence of fitness. These results indicate that Arrhenius
kinetics only apply over a narrow range of temperatures for ectotherms, complicating attempts to identify general patterns of temperature dependence.
Keywords: Arrhenius, fitness, temperature, universal temperature
dependence.
Introduction
Temperature affects all biological rate processes, from molecular kinetics to population growth. At the level of
whole-organism performance, rates of locomotion,
growth, development, and fitness have a predictable dependence on temperature. Performance increases gradually with increasing temperature, reaches a maximum at
the optimal temperature, Topt, and then declines rapidly
with further increases of temperature (Huey and Stevenson
1979; Huey and Hertz 1984; Huey and Kingsolver 1989).
This pattern is illustrated in figure 1 (top row) for the
* Corresponding author; e-mail: [email protected].
Am. Nat. 2010. Vol. 176, pp. 227–233. 䉷 2010 by The University of Chicago.
0003-0147/2010/17602-51851$15.00. All rights reserved.
DOI: 10.1086/653662
temperature dependence of fitness (intrinsic rate of population growth, r) for the bacteria Staphylococcus thermophilus and Pseudomonas fluorescens (modified from fig.
2 in Ratkowsky et al. 2005). The initial rise of fitness that
occurs with increasing temperature is attributed to Arrhenius kinetics, in which rates of reaction exponentially
increase with increasing temperature (Arrhenius 1915).
The decrease in fitness that occurs above Topt is attributed
to higher temperatures making proteins overly flexible,
reducing substrate affinity by disrupting the active site and
ultimately leading to denaturation of critical proteins
(Fields 2001; Hochachka and Somero 2002).
The Arrhenius equation has been used in many models
for rates of metabolism, growth, development, and fitness.
These models assume that these rates depend on ratelimiting reactions. Some models use the Arrhenius equation without modification (Gillooly et al. 2001; Savage et
al. 2004), while others modify the Arrhenius equation to
allow for thermal denaturation of the rate-limiting enzymes (Schoolfield et al. 1981; Ratkowsky et al. 2005).
Using only the Arrhenius equation (Arrhenius 1915), the
intrinsic rate of population growth (r) over temperatures
below Topt, the physiological temperature range (PTR), is
ln (r) p
⫺E
,
kT
(1)
where k is the Boltzmann constant, T is temperature (kelvin), and E is the activation energy parameter (in electron
volts [eV]) for the rate-limiting reactions. The Arrhenius
equation thus predicts a linear relationship between
ln (r) and (kT)⫺1 for temperatures within the PTR.
The Arrhenius model has recently been extended to
incorporate the scaling effects of body mass (M) on rates
of metabolism, individual growth, and fitness (Gillooly et
al. 2001; Savage et al. 2004). This model predicts the temperature dependence of mass-scaled fitness:
228 The American Naturalist
Figure 1: Representative microbial thermal reaction norms (adapted from fig. 2 in Ratkowsky et al. 2005). Top row, intrinsic growth rate of two
microbes as a function of temperature. Arrows indicate the optimal temperature of each microbe (respectively, 313 and 300 K). Temperature below
the optimal temperature (to the left of the arrow) would be considered part of the physiological temperature range. Bottom row, natural log of
growth rate as a function of 1/kT . The solid black line is the linear regression performed per the Arrhenius equation (eq. [1]). The linear slopes
are ⫺1.22 and ⫺0.66 eV for Staphylococcus thermophilus and Pseudomonas fluorescens, respectively.
ln (rM 1/4) p
⫺E
,
kT
(2)
where M is the mass of an individual in micrograms. Equation (2) predicts that mass-scaled population growth rates
(ln (rM 1/4)) are linearly dependent on inverse temperature
(kT)⫺1 with a slope given by ⫺E. A second prediction of
the model is that across diverse taxa, this activation energy
parameter, E, should have an average, or universal, value
of 0.6 eV. As a result, this model is sometimes called the
universal temperature dependence (UTD) model. These
predictions apply within some “physiologically relevant
temperature range” (PTR; Savage et al. 2004)—temperatures below the optimal temperature but excluding low
temperatures at which growth rate is negative.
A critical assumption of the Arrhenius and UTD models
is that the probability that the rate-limiting enzyme is in
its active conformation is 100% and that this does not
vary within the PTR (Schoolfield et al. 1981; Gillooly et
al. 2001; Ratkowsky et al. 2005). If the probability that the
rate-limiting reaction is in active form declines at high or
low temperatures within the PTR (Schoolfield et al. 1981;
Ratkowsky et al. 2005), then this will make the relationship
between ln (r) and (kT)⫺1 concave (curved downward)
rather than linear. As a result, there are two key empirical
questions to consider. Is the relationship between ln (r)
and inverse temperature (kT)⫺1 linear or concave within
the PTR? Is the slope of this relation constant across diverse
organisms, suggesting a single universal value for activation energy E (Savage et al. 2004)? The second question
Erroneous Arrhenius 229
has been explored empirically by several authors (Savage
et al. 2004; Frazier et al. 2006; Lopez-Urrutia and Moran
2007; Knies et al. 2009), with a thorough analysis of data
for metabolic and developmental rates in insects (Irlich et
al. 2009). The first question, along with its implications
for estimates of activation energy, has not been systematically explored. This is the goal of this note.
Previously, UTD-model predictions have been tested by
combining growth-rate data for multiple species or taxa,
each having intrinsic growth-rate measures at only a few
temperatures (e.g., Gillooly et al. 2001; Savage et al. 2004;
Irlich et al. 2009). However, the predictions of the UTD
model can best be assessed for individual species with
measurements of intrinsic growth-rate measures at many
temperatures below the optimal temperature (Topt). For
example, from Ratkowsky et al. (2005), S. thermophilus
and P. fluorescens have 13 and 42 growth-rate measures
below their Topt of 313 and 300 K, respectively (fig. 1). At
temperatures below the Topt of each species (within the
PTR), equation (1) predicts a negative and linear relationship between ln (r) and (kT)⫺1. A visual assessment
within the PTR suggests a relatively linear relationship for
P. fluorescens but a concave relationship for S. thermophilus
(fig. 1, bottom row). It is unclear which pattern is true for
most organisms. If most organisms do not show a linear
relationship between ln (r) and (kT)⫺1 for temperatures less
than Topt, it may be that this predicted linear relationship
still holds over a narrower range of temperatures.
Here we assess how well the Arrhenius model assumed
by UTD (eqq. [1], [2]) described data for intrinsic rate of
population growth (fitness, r) for individual species or
genotypes at temperatures within the physiological temperature range (PTR). We use two sources of data: microbes for which there are many measured temperatures
(Ratkowsky et al. 2005) and diverse plankton, algae, and
insect species for which there are relatively few measured
temperatures for each species (Savage et al. 2004). The
specific questions we focus on are (i) Is the temperature
dependence of (mass-scaled) fitness linear or concave below Topt? (ii) How might nonlinearity impact our estimates
of the slope? (iii) Can we define operationally a more
restricted temperature range within the PTR for which the
slope is linear? and (iv) Are there differences among species
in the slope within this PTR over which equation (1)
applies?
Methods
Data Sets
We consider data for clonal growth rate (r) for microbes
(Ratkowsky et al. 2005). This includes 35 data sets for nine
microbial species, each with growth-rate measures at many
temperatures (mean of 24) over a broad range (mean of
32.4⬚C). We restrict our analyses to temperatures within
the PTR (Savage et al. 2004) by excluding low temperatures
at which growth rate is negative and high temperatures
above the optimum temperature. To identify the optimum
temperature (Topt) in a consistent, quantitative, and unbiased manner, we first determine whether a third- or
fourth-order polynomial best describes the temperature
dependence of fitness of each data set, based on the smallsample Akaike Information Criterion (AIC; Burnham and
Anderson 2002). Then, Topt is defined for each data set as
the temperature at which r is maximized when evaluated
with the AIC best model (as shown in fig. 1, top row).
Because the thermal reaction norm is expected to become
nonlinear near Topt (Ratkowsky et al. 2005), we also choose
a narrower range of temperatures within the PTR. These
narrower temperature ranges are determined by identifying temperatures at which the growth rate is some fraction of the maximum growth rate (rmax), where rmax is the
growth rate at Topt.
We also reanalyze the data from Savage et al. (2004),
which have relatively few measurements (mean of four)
for each species. Savage et al. (2004) considered the UTD
model of intrinsic rate of population growth (mass-scaled
growth rate; eq. [2]) for unicellular organisms with nonoverlapping generations (unicellular algae and protists),
multicellular organisms with nonoverlapping generations
(plankton and insects), and multicellular organisms with
overlapping generations (vertebrates: mammals and fish).
Savage et al. (2004) excluded low temperatures at which
growth rate is negative and high temperatures at which
growth rate is clearly rapidly declining. Savage et al. (2004)
fitted equation (2) to each class of organisms and so estimated the activation energy for each class of organisms.
However, this approach obscures deviations of individual
species from the linear model predicted by equation (2).
Therefore, from the supplemental material in Savage et al.
(2004), we consider only species with growth-rate measures at four or more temperatures within the PTR, which
eliminated the data for all vertebrate species. This requirement ensured that we could statistically evaluate nonlinear
patterns between ln (r) and (kT)⫺1. After these exclusions,
there were 13 species with sufficient data for analysis, including five plankton species, three algae species, and five
insect species.
For each data set, we calculate the best-fit line and bestfit second-degree polynomial for the relationship between
ln (r) and 1/kT (eq. [1]) or that between ln (rM 1/4) and
1/kT (eq. [2]). For the Ratkowsky et al. (2005) analysis,
we used an F-test and the small-sample AIC (Burnham
and Anderson 2002) to determine whether the seconddegree polynomial is a significantly better fit than the linear
model. We report only the F-test results unless the tests
230 The American Naturalist
Table 1: Effect of temperature range on estimation of activation
energy for microbial species from Ratkowsky et al. (2005)
Mean
Mean
Mean
Linear
Temperature temperature number of
activation model
range
range (⬚C) temperatures energy (eV) best fit
PTR
.1–.9 of rmax
.2–.8 of rmax
.3–.7 of rmax
25.15
17.64
12.23
7.64
24.06
17.6
12.9
8.31
.89
.77
.75
.71
1/35
4/35
15/35
24/35
Note: PTR p physiological temperature range.
disagreed. For the Savage et al. (2004) analysis, the small
sample sizes precluded use of the small-sample AIC, and
so an F-test was used to determine the best-fit model. For
each data set, we also determined whether the best-fit
second-degree polynomial was concave up or concave
down by taking the second derivative of this function.
Permutations
To assess the effect of number of measurements on ability
to reject the linear model and to make the number of
measurements for each data set in Ratkowsky et al. (2005)
comparable to those in Savage et al. (2004), we also analyzed the Ratkowsky et al. (2005) data sets in another
way. For each Ratkowsky et al. (2005) data set with N
measurements in the PTR, we generated
range is 0.1–0.9 of rmax (mean of 17.6 measurements), then
the second-degree polynomial is a significantly better fit
to 91% of data sets. But this fraction drops to 57% (AIC:
54%) for 0.2–0.8 of rmax (mean of 12.9 measurements) and
to 26% (AIC: 16%, 5 of 31) when the temperature range
is restricted to temperatures at which r is 0.3–0.7 of rmax
(mean of 8.31 measurements). Over this restricted range,
the small sample size of four data sets does not allow the
use of AIC. Figure A1 in the online edition of the American
Naturalist depicts a representative comparison of the full
PTR and the temperature range at which r is 0.3–0.7 of
rmax for 10 data sets. At the temperature range (0.3–0.7 of
rmax) at which a majority (26 of 35) of data sets are fitted
better by the linear model, the average temperature range
is 7.6⬚C and the average number of measurements is 8.3
(as compared to 25.2⬚C and 24.1, respectively, for the average PTR). Across all data sets, the mean slope declines
from 0.89 for the full PTR to 0.71 when the temperature
range is restricted to 0.3–0.7 of rmax (fig. 2).
The change in the linear slope that accompanies a decreasing temperature range is not the same for all the
bacteria species. For six of nine species, including the three
species with the greatest number of data sets (Escherichia
( )
N⫺2
2
permutations, each of which maintained the original PTR
range, but we decreased the number of measurements to
four: one at the minimum of the PTR, one at the maximum of the PTR, and two at intervening temperatures.
For each permutation, we used an F-test, as above, to
determine whether the second-degree polynomial is a significantly better fit than the linear model. As in the Savage
et al. (2004) analysis, we could not use the small-sample
AIC here because of the small sample size.
Results
The linear Arrhenius model (eq. [1]) is a poor fit to the
35 data sets from Ratkowsky et al. (2005). Over the PTR
(mean of 25.2⬚C; range of 13.1⬚–32.9⬚C), the seconddegree polynomial is a significantly better fit (F-test, P !
.05) to 34 of 35 data sets (97%) and is concave down
(second-derivative ! 0) for all the data sets. As the temperature range examined was reduced, the number of data
sets best fitted by a second-degree polynomial became
smaller (table 1). For example, when the temperature
Figure 2: Linear slopes (activation energy estimates) as a function of
temperature range for microbial species. These data are for the species
from Ratkowsky et al. (2005). PTR represents the physiological temperature range and includes temperatures greater than 0⬚C and less than
Topt. For species where there was more than one data set (Escherichia coli,
Listeria monocytogenes, and Streptococcus thermophilus), error bars are
shown for the standard deviation in the slope estimate.
Erroneous Arrhenius 231
coli, Listeria monocytogenes, and Streptococcus thermophilus), the slope of linear fit decreases as temperature range
is narrowed. For three of nine species, the slope of linear
fit increases as temperature range is narrowed. These results suggest that restricting the temperature range within
the PTR causes the slope estimates to become more similar,
but it is unclear whether the estimates converge toward a
common value (fig. 2).
The number of temperatures at which growth rate is
measured impacts whether or not the linear Arrhenius
model is rejected. By restricting each data set from Ratkowsky et al. (2005) to just four temperatures and sampling all possible four-temperature samples for each data
set, each data set rejects the linear model in an average of
only 25% (range: 1.5%–41%) of its permutations.
The Savage et al. (2004) data have fewer measurements
(range: four to seven) for each species, making it difficult
to assess the fit of the linear UTD model (eq. [2]; fig. 3).
For 11 of the 13 species, the best-fit second-degree polynomial is concave down (second derivative ! 0). However,
for only four of the 13 species is the second-degree polynomial a significantly better fit (F-test, P ! .05) than a
linear model for the dependence of ln (rM 1/4) on (kT)⫺1.
Discussion
Microbiologists and entomologists have long recognized
the limitations of applying the Arrhenius equation to data
for growth and developmental rates. Ratkowsky et al.
(1982) pointed out that population growth rates for microbes were concave down rather than linear when plotted
in the Arrhenius manner. We found that significant negative curvature—concavity—occurs in most data sets for
temperature below Topt for the nine microbial species from
Ratkowsky et al. (2005). We found that concavity also
occurs in the majority of species from Savage et al. (2004),
including plankton, insect, and algae species, though significant concavity was detected in only four of the 13
species, presumably due to lack of statistical power in these
data sets.
The linear relation predicted by Arrhenius-based models
for growth rate as a function of inverse temperature is
true only when 100% of the rate-limiting enzyme is in an
active form. The UTD model prediction (eq. [2]) follows
Arrhenius in assuming that the probability of the enzyme
being in the active form is high (∼1) and constant over
some temperature range (the PTR). The concavity that we
Figure 3: Linear UTD model fitted to diverse organisms. The three types of organisms in Savage et al. (2004) are shown. A–C, Best-fit line to each
species and the species data points. The best-fit line to the combined species data is shown in black. D–F, Best-fit polynomial of degree 2 for each
species. The best-fit line to the combined species data is shown in black.
232 The American Naturalist
observe violates these model predictions and can be accounted for by incorporating a probability of activation
as a function of temperature (Schoolfield et al. 1981; Ratkowsky et al. 2005). Models that estimate the probability
of activity as a function of temperature for rates of growth
or differentiation show that the fraction of active enzyme
is not 100% and is not constant at temperatures below
Topt. Rather, these models find support for enzyme inactivation occurring at the lower and upper ends of the PTR,
and it is only in the intermediate range of the PTR that
enzyme activation is at or near 100% (Schoolfield et al.
1981; van der Have and de Jong 1996; Ratkowsky et al.
2005).
Other factors may also contribute to the nonlinearity
observed here. If the Arrhenius model is a purely probabilistic description of reaction kinetics, then one might
expect increasing variation in growth rate toward the extremes of any line fitted through the data, generating random variation in the degree and direction of curvature.
However, we detected negative concavity in all microbial
data sets and in 11 of 13 of the Savage et al. (2004) data
sets; in no case did we detect significant upward concavity.
This pattern is inconsistent with a probabilistic interpretation of Arrhenius. Second, the rate-limiting enzyme may
not be saturated by substrate across all temperatures in
the PTR. To produce the downward concavity that we see
in nearly all the data sets, the probability of rate-limiting
enzymes being saturated would have to decline toward the
extremes of the PTR. We are not aware of data that address
this hypothesis.
The assumption that Arrhenius governs growth rates
below Topt is particularly problematic when, as in most
cases, only a handful of temperatures are evaluated (Savage
et al. 2004). In this case, it is not clear what is being
estimated by the activation energy (slope) term used to
describe the temperature sensitivity of growth rate of different groups. It is also not clear that there is a single
common slope (Terblanche et al. 2007). One potential
solution to this problem of concavity is to define a restricted range of temperatures below Topt within which the
linear model holds. But identifying this range operationally
is a challenge and still requires measurements at many
temperatures. Our results clearly show that the choice of
the temperature range can strongly affect the estimated
slope (fig. 2; table 1). We find that the linear model is
generally better than the concave model only over a very
narrow range within the PTR—8⬚C on average—limiting
the generality of this model. Even when a restricted range
for PTR is used (in which linearity applies), there is substantial variation in estimated slopes (table 1: 0.5–0.9 eV)
among species of microbes. They do cluster around the
predicted value of 0.6–0.7 eV. This may indicate that there
is a universal temperature dependence, but only over a
narrow range of a species PTR because of high- and lowtemperature inactivation of critical enzymes. We are unable to eliminate the possibility that at the narrowest temperature range (0.3–0.7 of rmax), the linear model is not
rejected by many of the data sets because the number of
measurements is too few (mean of 8.31, range of 4–23).
Evidence from enzyme and physiological studies also
challenges the assumption of a universal E based on Arrhenius. Studies of enzyme kinetics suggest that enzymes
from cold-adapted species tend to have a lower E than
orthologs from warm-adapted species. This reduction in
activation energy for cold-adapted enzymes will increase
rates of reaction and may be one mechanism of adaptation
to cold temperatures (Feller and Gerday 1997; Lonhienne
et al. 2000; Fields et al. 2002; Feller 2003). For multicellular
animals, metabolic fluxes may be limited by oxygen supply
such that no single enzyme is rate limiting across the entire
physiological temperature range, altering the temperature
dependence of organismal rate processes (Pörtner et al.
2006).
Acknowledgments
D. Ratkowsky generously shared data from Ratkowsky et
al. (2005). The Weinreich lab and R. Early provided insightful comments about the analyses and final version.
We thank J. Terblanche and an anonymous reviewer for
their thoughtful comments. A. R. Hitch provided title inspiration. Research was supported in part by National Science Foundation grant IOS 0641179 to J.G.K. and National
Institutes of Health grant F32 GM086105-01A1 to J.L.K.
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Associate Editor: Ary A. Hoffmann
Editor: Mark A. McPeek
Left, “embryo of a Corophium, magnified ninety diameters; the mouth-parts are similar to the legs in form. The yolk mass (y) lies on the back of
the animal; h is the head, and m the mouth-parts.” Right, “the larva, or ‘Nauplius,’ of a shrimp, magnified forty-five diameters. The body is soft,
oval, in form somewhat like a mite.” From “The Metamorphosis of Crabs: A Review of Für Darwin, Fritz Müller, Leipzig, 1864” (American Naturalist,
1869, 3:432–435).